SingletonsAndAntisingletons

Singleton blocks plus antisingletons.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000249via SetPartition✓↗

SingletonsAndAntisingletons(partition)Singleton blocks plus antisingletons.

Typed signatures

Type	Package	Applies
(set_partition) -> number	combinatorics	always

Domain: Set partition statistics

Details

• Defined over SetPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a SetPartition and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

0 if Length(_x) < 1 else Count(Filter(_x, Function(Length(b) == 1, b))) +
                         (0 if Length(_x) < 1 else (0 if Length(Flatten(Map(
                                                                             Function(Most(b), b),
                                                                             _x,
                                                                           ))) <
                                                         1 else Count(Filter(
                                                                             Range(
                                                                                    1,
                                                                                    Length(Flatten(Map(Function(Most(b), b), _x))),
                                                                                    1,
                                                                                  ),
                                                                             Function(Flatten(Map(Function(Rest(b), b), _x))[i] -
                                                                                      Flatten(Map(Function(Most(b), b), _x))[i] ==
                                                                                      1, i),
                                                                           ))) +
                                                   (1 if Last(_x[1]) == Sum(Map(
                                                                                 Function(Length(b), b),
                                                                                 _x,
                                                                               )) else 0))

Examples

Basic #

the partition $\{1,3\} \mid \{2\}$

live ↗

In$\mathrm{SingletonsAndAntisingletons}({\operatorname{SetPartition}}([[1, 3], [2]]))$Out$2$

the partition $\{1\} \mid \{2,4\} \mid \{3\}$

live ↗

In$\mathrm{SingletonsAndAntisingletons}({\operatorname{SetPartition}}([[1], [2, 4], [3]]))$Out$2$

Properties #

FindStat's value for {{1,2}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2]]), \text{SingletonsAndAntisingletons})$Out$2$

Known value · FindStat St000249

FindStat's value for {{1,2,3}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2, 3]]), \text{SingletonsAndAntisingletons})$Out$3$

Known value · FindStat St000249

FindStat's value for {{1},{2}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1], [2]]), \text{SingletonsAndAntisingletons})$Out$2$

Known value · FindStat St000249

About

Examples	5
Crosswalk	FindStat